Simplify square root of 3^16

Solution:

Step 1:

Compute the value of $3^{16}$ to find the number under the square root. $\sqrt{43046721}$

Step 2:

Express the number $43046721$ as a perfect square, which is $\left(6561\right)^{2}$. $\sqrt{\left(6561\right)^{2}}$

Step 3:

Extract the square root by removing the radical sign, considering only the positive root as per convention. $6561$

Knowledge Notes:

To simplify the square root of a number, especially when the number is a power of an integer, we can use the properties of exponents and radicals. Here are some relevant knowledge points:

1. Exponentiation: Raising a number to a power is a form of exponentiation. For example, $3^{16}$ means multiplying $3$ by itself $15$ more times.

2. Perfect Squares: A perfect square is a number that can be expressed as the square of an integer. In this case, $6561$ is a perfect square because it is equal to $81^2$.

3. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of a perfect square is always an integer.

4. Radical Simplification: When simplifying radicals, if the number under the radical is a perfect square, the square root is simply the integer whose square is the radicand (the number under the radical).

5. Positive Real Numbers: When taking the square root of a number, we usually consider the principal (positive) square root because in most contexts, we are interested in real numbers.

6. Notation: The radical symbol $\sqrt{}$ denotes the square root, and the number inside the symbol is called the radicand.

7. Latex Formatting: In Latex, exponents are written using the caret symbol $^$, and the square root is written using the command $\sqrt{}$.

By understanding these concepts, one can simplify square roots of large numbers, especially when they are powers of integers, by breaking them down into perfect squares and then taking the square root.